prime_power_valuation_square_even

Exact expanded first-order arithmetic statement

forall p a e E. ((~(p = 1) /\ forall frm_prime_left_ftsv_prime frm_prime_right_ftsv_prime. p = frm_prime_left_ftsv_prime * frm_prime_right_ftsv_prime -> frm_prime_left_ftsv_prime = 1 \/ frm_prime_right_ftsv_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_ftsv_square_input_exponent_bound. bpv_gap_ftsv_square_input_exponent_bound + e = a) /\ (exists bpv_result_ftsv_square_input_selected. ((exists ff_b_ftsv_square_input_selected_power ff_c_ftsv_square_input_selected_power. ((forall ff_i_ftsv_square_input_selected_power_repeat. (exists ff_lt_ftsv_square_input_selected_power_repeat_bound. ff_lt_ftsv_square_input_selected_power_repeat_bound + S ff_i_ftsv_square_input_selected_power_repeat = e) -> (((exists ff_h_ftsv_square_input_selected_power_repeat_decoded. ff_h_ftsv_square_input_selected_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_square_input_selected_power_repeat)) * ff_c_ftsv_square_input_selected_power)) /\ exists ff_q_ftsv_square_input_selected_power_repeat_decoded. ff_b_ftsv_square_input_selected_power = ff_q_ftsv_square_input_selected_power_repeat_decoded * S ((S (ff_i_ftsv_square_input_selected_power_repeat)) * ff_c_ftsv_square_input_selected_power) + (p)))) /\ (exists ff_u_ftsv_square_input_selected_power_product ff_v_ftsv_square_input_selected_power_product. ((((exists ff_h_ftsv_square_input_selected_power_product_start. ff_h_ftsv_square_input_selected_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_square_input_selected_power_product)) /\ exists ff_q_ftsv_square_input_selected_power_product_start. ff_u_ftsv_square_input_selected_power_product = ff_q_ftsv_square_input_selected_power_product_start * S ((S (0)) * ff_v_ftsv_square_input_selected_power_product) + (1))) /\ ((((exists ff_h_ftsv_square_input_selected_power_product_terminal. ff_h_ftsv_square_input_selected_power_product_terminal + S (bpv_result_ftsv_square_input_selected) = S ((S (e)) * ff_v_ftsv_square_input_selected_power_product)) /\ exists ff_q_ftsv_square_input_selected_power_product_terminal. ff_u_ftsv_square_input_selected_power_product = ff_q_ftsv_square_input_selected_power_product_terminal * S ((S (e)) * ff_v_ftsv_square_input_selected_power_product) + (bpv_result_ftsv_square_input_selected))) /\ forall ff_i_ftsv_square_input_selected_power_product. (exists ff_lt_ftsv_square_input_selected_power_product_bound. ff_lt_ftsv_square_input_selected_power_product_bound + S ff_i_ftsv_square_input_selected_power_product = e) -> exists ff_p_ftsv_square_input_selected_power_product ff_r_ftsv_square_input_selected_power_product ff_s_ftsv_square_input_selected_power_product. ((((exists ff_h_ftsv_square_input_selected_power_product_factor. ff_h_ftsv_square_input_selected_power_product_factor + S (ff_p_ftsv_square_input_selected_power_product) = S ((S (ff_i_ftsv_square_input_selected_power_product)) * ff_c_ftsv_square_input_selected_power)) /\ exists ff_q_ftsv_square_input_selected_power_product_factor. ff_b_ftsv_square_input_selected_power = ff_q_ftsv_square_input_selected_power_product_factor * S ((S (ff_i_ftsv_square_input_selected_power_product)) * ff_c_ftsv_square_input_selected_power) + (ff_p_ftsv_square_input_selected_power_product))) /\ ((((exists ff_h_ftsv_square_input_selected_power_product_partial. ff_h_ftsv_square_input_selected_power_product_partial + S (ff_r_ftsv_square_input_selected_power_product) = S ((S (ff_i_ftsv_square_input_selected_power_product)) * ff_v_ftsv_square_input_selected_power_product)) /\ exists ff_q_ftsv_square_input_selected_power_product_partial. ff_u_ftsv_square_input_selected_power_product = ff_q_ftsv_square_input_selected_power_product_partial * S ((S (ff_i_ftsv_square_input_selected_power_product)) * ff_v_ftsv_square_input_selected_power_product) + (ff_r_ftsv_square_input_selected_power_product))) /\ ((((exists ff_h_ftsv_square_input_selected_power_product_successor. ff_h_ftsv_square_input_selected_power_product_successor + S (ff_s_ftsv_square_input_selected_power_product) = S ((S (S ff_i_ftsv_square_input_selected_power_product)) * ff_v_ftsv_square_input_selected_power_product)) /\ exists ff_q_ftsv_square_input_selected_power_product_successor. ff_u_ftsv_square_input_selected_power_product = ff_q_ftsv_square_input_selected_power_product_successor * S ((S (S ff_i_ftsv_square_input_selected_power_product)) * ff_v_ftsv_square_input_selected_power_product) + (ff_s_ftsv_square_input_selected_power_product))) /\ ff_s_ftsv_square_input_selected_power_product = ff_r_ftsv_square_input_selected_power_product * ff_p_ftsv_square_input_selected_power_product)))))))) /\ (exists bpv_factor_ftsv_square_input_selected_divides. a = bpv_result_ftsv_square_input_selected * bpv_factor_ftsv_square_input_selected_divides)))) /\ forall bpv_candidate_ftsv_square_input. (exists bpv_gap_ftsv_square_input_candidate_bound. bpv_gap_ftsv_square_input_candidate_bound + bpv_candidate_ftsv_square_input = a) -> (exists bpv_result_ftsv_square_input_candidate. ((exists ff_b_ftsv_square_input_candidate_power ff_c_ftsv_square_input_candidate_power. ((forall ff_i_ftsv_square_input_candidate_power_repeat. (exists ff_lt_ftsv_square_input_candidate_power_repeat_bound. ff_lt_ftsv_square_input_candidate_power_repeat_bound + S ff_i_ftsv_square_input_candidate_power_repeat = bpv_candidate_ftsv_square_input) -> (((exists ff_h_ftsv_square_input_candidate_power_repeat_decoded. ff_h_ftsv_square_input_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_square_input_candidate_power_repeat)) * ff_c_ftsv_square_input_candidate_power)) /\ exists ff_q_ftsv_square_input_candidate_power_repeat_decoded. ff_b_ftsv_square_input_candidate_power = ff_q_ftsv_square_input_candidate_power_repeat_decoded * S ((S (ff_i_ftsv_square_input_candidate_power_repeat)) * ff_c_ftsv_square_input_candidate_power) + (p)))) /\ (exists ff_u_ftsv_square_input_candidate_power_product ff_v_ftsv_square_input_candidate_power_product. ((((exists ff_h_ftsv_square_input_candidate_power_product_start. ff_h_ftsv_square_input_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_square_input_candidate_power_product)) /\ exists ff_q_ftsv_square_input_candidate_power_product_start. ff_u_ftsv_square_input_candidate_power_product = ff_q_ftsv_square_input_candidate_power_product_start * S ((S (0)) * ff_v_ftsv_square_input_candidate_power_product) + (1))) /\ ((((exists ff_h_ftsv_square_input_candidate_power_product_terminal. ff_h_ftsv_square_input_candidate_power_product_terminal + S (bpv_result_ftsv_square_input_candidate) = S ((S (bpv_candidate_ftsv_square_input)) * ff_v_ftsv_square_input_candidate_power_product)) /\ exists ff_q_ftsv_square_input_candidate_power_product_terminal. ff_u_ftsv_square_input_candidate_power_product = ff_q_ftsv_square_input_candidate_power_product_terminal * S ((S (bpv_candidate_ftsv_square_input)) * ff_v_ftsv_square_input_candidate_power_product) + (bpv_result_ftsv_square_input_candidate))) /\ forall ff_i_ftsv_square_input_candidate_power_product. (exists ff_lt_ftsv_square_input_candidate_power_product_bound. ff_lt_ftsv_square_input_candidate_power_product_bound + S ff_i_ftsv_square_input_candidate_power_product = bpv_candidate_ftsv_square_input) -> exists ff_p_ftsv_square_input_candidate_power_product ff_r_ftsv_square_input_candidate_power_product ff_s_ftsv_square_input_candidate_power_product. ((((exists ff_h_ftsv_square_input_candidate_power_product_factor. ff_h_ftsv_square_input_candidate_power_product_factor + S (ff_p_ftsv_square_input_candidate_power_product) = S ((S (ff_i_ftsv_square_input_candidate_power_product)) * ff_c_ftsv_square_input_candidate_power)) /\ exists ff_q_ftsv_square_input_candidate_power_product_factor. ff_b_ftsv_square_input_candidate_power = ff_q_ftsv_square_input_candidate_power_product_factor * S ((S (ff_i_ftsv_square_input_candidate_power_product)) * ff_c_ftsv_square_input_candidate_power) + (ff_p_ftsv_square_input_candidate_power_product))) /\ ((((exists ff_h_ftsv_square_input_candidate_power_product_partial. ff_h_ftsv_square_input_candidate_power_product_partial + S (ff_r_ftsv_square_input_candidate_power_product) = S ((S (ff_i_ftsv_square_input_candidate_power_product)) * ff_v_ftsv_square_input_candidate_power_product)) /\ exists ff_q_ftsv_square_input_candidate_power_product_partial. ff_u_ftsv_square_input_candidate_power_product = ff_q_ftsv_square_input_candidate_power_product_partial * S ((S (ff_i_ftsv_square_input_candidate_power_product)) * ff_v_ftsv_square_input_candidate_power_product) + (ff_r_ftsv_square_input_candidate_power_product))) /\ ((((exists ff_h_ftsv_square_input_candidate_power_product_successor. ff_h_ftsv_square_input_candidate_power_product_successor + S (ff_s_ftsv_square_input_candidate_power_product) = S ((S (S ff_i_ftsv_square_input_candidate_power_product)) * ff_v_ftsv_square_input_candidate_power_product)) /\ exists ff_q_ftsv_square_input_candidate_power_product_successor. ff_u_ftsv_square_input_candidate_power_product = ff_q_ftsv_square_input_candidate_power_product_successor * S ((S (S ff_i_ftsv_square_input_candidate_power_product)) * ff_v_ftsv_square_input_candidate_power_product) + (ff_s_ftsv_square_input_candidate_power_product))) /\ ff_s_ftsv_square_input_candidate_power_product = ff_r_ftsv_square_input_candidate_power_product * ff_p_ftsv_square_input_candidate_power_product)))))))) /\ (exists bpv_factor_ftsv_square_input_candidate_divides. a = bpv_result_ftsv_square_input_candidate * bpv_factor_ftsv_square_input_candidate_divides))) -> (exists bpv_gap_ftsv_square_input_maximal. bpv_gap_ftsv_square_input_maximal + bpv_candidate_ftsv_square_input = e)) -> (((exists bpv_gap_ftsv_square_output_exponent_bound. bpv_gap_ftsv_square_output_exponent_bound + E = (a * a)) /\ (exists bpv_result_ftsv_square_output_selected. ((exists ff_b_ftsv_square_output_selected_power ff_c_ftsv_square_output_selected_power. ((forall ff_i_ftsv_square_output_selected_power_repeat. (exists ff_lt_ftsv_square_output_selected_power_repeat_bound. ff_lt_ftsv_square_output_selected_power_repeat_bound + S ff_i_ftsv_square_output_selected_power_repeat = E) -> (((exists ff_h_ftsv_square_output_selected_power_repeat_decoded. ff_h_ftsv_square_output_selected_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_square_output_selected_power_repeat)) * ff_c_ftsv_square_output_selected_power)) /\ exists ff_q_ftsv_square_output_selected_power_repeat_decoded. ff_b_ftsv_square_output_selected_power = ff_q_ftsv_square_output_selected_power_repeat_decoded * S ((S (ff_i_ftsv_square_output_selected_power_repeat)) * ff_c_ftsv_square_output_selected_power) + (p)))) /\ (exists ff_u_ftsv_square_output_selected_power_product ff_v_ftsv_square_output_selected_power_product. ((((exists ff_h_ftsv_square_output_selected_power_product_start. ff_h_ftsv_square_output_selected_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_square_output_selected_power_product)) /\ exists ff_q_ftsv_square_output_selected_power_product_start. ff_u_ftsv_square_output_selected_power_product = ff_q_ftsv_square_output_selected_power_product_start * S ((S (0)) * ff_v_ftsv_square_output_selected_power_product) + (1))) /\ ((((exists ff_h_ftsv_square_output_selected_power_product_terminal. ff_h_ftsv_square_output_selected_power_product_terminal + S (bpv_result_ftsv_square_output_selected) = S ((S (E)) * ff_v_ftsv_square_output_selected_power_product)) /\ exists ff_q_ftsv_square_output_selected_power_product_terminal. ff_u_ftsv_square_output_selected_power_product = ff_q_ftsv_square_output_selected_power_product_terminal * S ((S (E)) * ff_v_ftsv_square_output_selected_power_product) + (bpv_result_ftsv_square_output_selected))) /\ forall ff_i_ftsv_square_output_selected_power_product. (exists ff_lt_ftsv_square_output_selected_power_product_bound. ff_lt_ftsv_square_output_selected_power_product_bound + S ff_i_ftsv_square_output_selected_power_product = E) -> exists ff_p_ftsv_square_output_selected_power_product ff_r_ftsv_square_output_selected_power_product ff_s_ftsv_square_output_selected_power_product. ((((exists ff_h_ftsv_square_output_selected_power_product_factor. ff_h_ftsv_square_output_selected_power_product_factor + S (ff_p_ftsv_square_output_selected_power_product) = S ((S (ff_i_ftsv_square_output_selected_power_product)) * ff_c_ftsv_square_output_selected_power)) /\ exists ff_q_ftsv_square_output_selected_power_product_factor. ff_b_ftsv_square_output_selected_power = ff_q_ftsv_square_output_selected_power_product_factor * S ((S (ff_i_ftsv_square_output_selected_power_product)) * ff_c_ftsv_square_output_selected_power) + (ff_p_ftsv_square_output_selected_power_product))) /\ ((((exists ff_h_ftsv_square_output_selected_power_product_partial. ff_h_ftsv_square_output_selected_power_product_partial + S (ff_r_ftsv_square_output_selected_power_product) = S ((S (ff_i_ftsv_square_output_selected_power_product)) * ff_v_ftsv_square_output_selected_power_product)) /\ exists ff_q_ftsv_square_output_selected_power_product_partial. ff_u_ftsv_square_output_selected_power_product = ff_q_ftsv_square_output_selected_power_product_partial * S ((S (ff_i_ftsv_square_output_selected_power_product)) * ff_v_ftsv_square_output_selected_power_product) + (ff_r_ftsv_square_output_selected_power_product))) /\ ((((exists ff_h_ftsv_square_output_selected_power_product_successor. ff_h_ftsv_square_output_selected_power_product_successor + S (ff_s_ftsv_square_output_selected_power_product) = S ((S (S ff_i_ftsv_square_output_selected_power_product)) * ff_v_ftsv_square_output_selected_power_product)) /\ exists ff_q_ftsv_square_output_selected_power_product_successor. ff_u_ftsv_square_output_selected_power_product = ff_q_ftsv_square_output_selected_power_product_successor * S ((S (S ff_i_ftsv_square_output_selected_power_product)) * ff_v_ftsv_square_output_selected_power_product) + (ff_s_ftsv_square_output_selected_power_product))) /\ ff_s_ftsv_square_output_selected_power_product = ff_r_ftsv_square_output_selected_power_product * ff_p_ftsv_square_output_selected_power_product)))))))) /\ (exists bpv_factor_ftsv_square_output_selected_divides. (a * a) = bpv_result_ftsv_square_output_selected * bpv_factor_ftsv_square_output_selected_divides)))) /\ forall bpv_candidate_ftsv_square_output. (exists bpv_gap_ftsv_square_output_candidate_bound. bpv_gap_ftsv_square_output_candidate_bound + bpv_candidate_ftsv_square_output = (a * a)) -> (exists bpv_result_ftsv_square_output_candidate. ((exists ff_b_ftsv_square_output_candidate_power ff_c_ftsv_square_output_candidate_power. ((forall ff_i_ftsv_square_output_candidate_power_repeat. (exists ff_lt_ftsv_square_output_candidate_power_repeat_bound. ff_lt_ftsv_square_output_candidate_power_repeat_bound + S ff_i_ftsv_square_output_candidate_power_repeat = bpv_candidate_ftsv_square_output) -> (((exists ff_h_ftsv_square_output_candidate_power_repeat_decoded. ff_h_ftsv_square_output_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_square_output_candidate_power_repeat)) * ff_c_ftsv_square_output_candidate_power)) /\ exists ff_q_ftsv_square_output_candidate_power_repeat_decoded. ff_b_ftsv_square_output_candidate_power = ff_q_ftsv_square_output_candidate_power_repeat_decoded * S ((S (ff_i_ftsv_square_output_candidate_power_repeat)) * ff_c_ftsv_square_output_candidate_power) + (p)))) /\ (exists ff_u_ftsv_square_output_candidate_power_product ff_v_ftsv_square_output_candidate_power_product. ((((exists ff_h_ftsv_square_output_candidate_power_product_start. ff_h_ftsv_square_output_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_square_output_candidate_power_product)) /\ exists ff_q_ftsv_square_output_candidate_power_product_start. ff_u_ftsv_square_output_candidate_power_product = ff_q_ftsv_square_output_candidate_power_product_start * S ((S (0)) * ff_v_ftsv_square_output_candidate_power_product) + (1))) /\ ((((exists ff_h_ftsv_square_output_candidate_power_product_terminal. ff_h_ftsv_square_output_candidate_power_product_terminal + S (bpv_result_ftsv_square_output_candidate) = S ((S (bpv_candidate_ftsv_square_output)) * ff_v_ftsv_square_output_candidate_power_product)) /\ exists ff_q_ftsv_square_output_candidate_power_product_terminal. ff_u_ftsv_square_output_candidate_power_product = ff_q_ftsv_square_output_candidate_power_product_terminal * S ((S (bpv_candidate_ftsv_square_output)) * ff_v_ftsv_square_output_candidate_power_product) + (bpv_result_ftsv_square_output_candidate))) /\ forall ff_i_ftsv_square_output_candidate_power_product. (exists ff_lt_ftsv_square_output_candidate_power_product_bound. ff_lt_ftsv_square_output_candidate_power_product_bound + S ff_i_ftsv_square_output_candidate_power_product = bpv_candidate_ftsv_square_output) -> exists ff_p_ftsv_square_output_candidate_power_product ff_r_ftsv_square_output_candidate_power_product ff_s_ftsv_square_output_candidate_power_product. ((((exists ff_h_ftsv_square_output_candidate_power_product_factor. ff_h_ftsv_square_output_candidate_power_product_factor + S (ff_p_ftsv_square_output_candidate_power_product) = S ((S (ff_i_ftsv_square_output_candidate_power_product)) * ff_c_ftsv_square_output_candidate_power)) /\ exists ff_q_ftsv_square_output_candidate_power_product_factor. ff_b_ftsv_square_output_candidate_power = ff_q_ftsv_square_output_candidate_power_product_factor * S ((S (ff_i_ftsv_square_output_candidate_power_product)) * ff_c_ftsv_square_output_candidate_power) + (ff_p_ftsv_square_output_candidate_power_product))) /\ ((((exists ff_h_ftsv_square_output_candidate_power_product_partial. ff_h_ftsv_square_output_candidate_power_product_partial + S (ff_r_ftsv_square_output_candidate_power_product) = S ((S (ff_i_ftsv_square_output_candidate_power_product)) * ff_v_ftsv_square_output_candidate_power_product)) /\ exists ff_q_ftsv_square_output_candidate_power_product_partial. ff_u_ftsv_square_output_candidate_power_product = ff_q_ftsv_square_output_candidate_power_product_partial * S ((S (ff_i_ftsv_square_output_candidate_power_product)) * ff_v_ftsv_square_output_candidate_power_product) + (ff_r_ftsv_square_output_candidate_power_product))) /\ ((((exists ff_h_ftsv_square_output_candidate_power_product_successor. ff_h_ftsv_square_output_candidate_power_product_successor + S (ff_s_ftsv_square_output_candidate_power_product) = S ((S (S ff_i_ftsv_square_output_candidate_power_product)) * ff_v_ftsv_square_output_candidate_power_product)) /\ exists ff_q_ftsv_square_output_candidate_power_product_successor. ff_u_ftsv_square_output_candidate_power_product = ff_q_ftsv_square_output_candidate_power_product_successor * S ((S (S ff_i_ftsv_square_output_candidate_power_product)) * ff_v_ftsv_square_output_candidate_power_product) + (ff_s_ftsv_square_output_candidate_power_product))) /\ ff_s_ftsv_square_output_candidate_power_product = ff_r_ftsv_square_output_candidate_power_product * ff_p_ftsv_square_output_candidate_power_product)))))))) /\ (exists bpv_factor_ftsv_square_output_candidate_divides. (a * a) = bpv_result_ftsv_square_output_candidate * bpv_factor_ftsv_square_output_candidate_divides))) -> (exists bpv_gap_ftsv_square_output_maximal. bpv_gap_ftsv_square_output_maximal + bpv_candidate_ftsv_square_output = E)) -> E = e + e